Let $K$ denote the kernel of the group homomorphism $f : G \rightarrow H$. Let $i : K \rightarrow G$ denote the inclusion homomorphism.
Then evidently $(K, i)$ is a universal arrow in the comma category $(S \downarrow G)$ for some functor $S$.
Question: How so? What is this $S$?
Attempt: I know that if $T$ is another group and $j$ another mapping that goes from $T$ to $G$ with the property that $j(T) \in K$, then there exists a unique mapping $j' : T \rightarrow K$ s.t. $i \circ j' = j$. Namely, $j'$ is just $j$ with codomain narrowed to $K$.